Question

Show that the following set of functions are subspaces of F(-infinity,infinity)

a) all everywhere differentiable function that satisfy f'+2f=0 ...?

Answer 1

Let suppose there are two functions that satisfy the equations
f ' +2f =0
and g' + 2g =0

If we add both of them
f' +g' + 2f + 2g =0
Grouping both the sides
(f+g) ' + 2 (f +g) =0So, if both g and f satisfies it then their sum also satisfy it, So there is a subspace and their sum satisfy them too.